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BUBBLING ABOVE THE THRESHOLD OF THE
SCALAR CURVATURE IN DIMENSIONS FOUR
AND FIVE
Bruno Premoselli, Pierre-Damien Thizy
To cite this version:
Bruno Premoselli, Pierre-Damien Thizy. BUBBLING ABOVE THE THRESHOLD OF THE
CURVATURE IN DIMENSIONS FOUR AND FIVE.
BRUNO PREMOSELLI AND PIERRE-DAMIEN THIZY
Abstract. On any closed manifold (Mn, g) of dimension n ∈ {4, 5} we exhibit new blow-up configurations for perturbations of a purely critical stationary Schr¨odinger equation. We construct positive solutions which blow-up as the sum of two isolated bubbles, one of which concentrates at a point ξ where the potential k of the equation satisfies
k(ξ) > n − 2 4(n − 1)Sg(ξ),
where Sg is the scalar curvature of (Mn, g). The latter condition requires the bubbles
to blow-up at different speeds and forces us to work at an elevated precision. We take care of this by performing a construction which combines a priori asymptotic analysis methods with a Lyapounov-Schmidt reduction.
1. Introduction
1.1. Statement of the results. Let (M, g) be a smooth closed Riemannian manifold of dimension n ≥ 3. Let 4g = −divg(∇·) be the Laplace-Beltrami operator and let k be
a smooth function in M such that 4g+ k is coercive. We are interested in this paper
in the existence of energy-bounded blowing-up families of positive solutions (uε)ε>0 to
critical stationary Schr¨odinger equations of the following type:
4guε+ kεuε= u2
∗−1
ε in M, (1.1)
where 2∗ = n−22n is the critical power for the embedding of H1(M ) into Lebesgue spaces and (kε)ε>0 is a smooth perturbation of k. We say that a family (uε)ε>0 of solutions of
(1.1) has bounded energy if
lim sup
ε→0
kuεkH1(M )< +∞.
Since the work of Struwe [26] it is known that if (uε)ε>0 has bounded energy then, up to
a subsequence, there exist k ∈ N, k sequences (µ1,ε)ε, . . . , (µk,ε)ε of positive real numbers
converging to zero and k sequences (ξ1,ε)ε, . . . , (ξk,ε)ε of points of M such that
uε= u0+ k
X
i=1
Wi,ε+ o(1) in H1(M ), (1.2)
The first author is supported by a FNRS grant MIS F.4522.15.
where the Wi,ε are bubbling profiles given by Wi,ε= µi,ε µ2 i,ε+ dg(ξi,ε,·)2 n(n−2) n−2 2 , (1.3)
dg is the geodesic distance and uε * u0 as ε → 0. We say that the family (uε)ε>0 of
solutions of (1.1) blows-up if
lim sup
ε→0
kuεkC0(M ) = +∞.
If (uε)ε>0 has bounded energy and blows-up it is easily seen that k ≥ 1 in (1.2).
In the last decades, a vast amount of work was poured into understanding when equations (1.1) possess blowing-up families of positive solutions – with and without the
energy-bound assumption. It turns out that the geometric potential k ≡ cnSg plays a threshold
role, where we have let cn = 4(n−1)n−2 and where Sg denotes the scalar curvature of
(M, g). It was indeed proven in [6] that when n ≥ 4 (1.1) has no blowing-up positive solutions whatsoever if k < cnSg, and in [5] that (1.1) has no energy-bounded
blowing-up positive solutions if k > cnSg (unless maybe if n = 6, see [5]). When n = 3 the
situation is completely different, see [9]. The latter result is based on the generalization of decomposition (1.2) to C0(M ) obtained in [7]. On the other side, energy-bounded
blowing-up families of positive solutions of (1.1) have been constructed when kε is a
small perturbation of cnSg, see for instance [8, 17, 25]. In another direction, if kε≡ cnSg
for all ε, equation (1.1) is the Yamabe equation, and its compactness properties exhibit intriguing dimensional phenomena, see [2, 5, 11, 13, 14, 15].
In this article we construct, in dimensions four and five, exotic bubbling configurations for (1.1), where the potential k lies well above the threshold of the scalar curvature at one of the concentration points. In particular, our equations are not perturbations of the Yamabe equation. Let (M, g) be a n-dimensional closed Riemannian manifold, n ∈ {4, 5}, and let Ψ ∈ Cc∞(Rn) be a smooth compactly supported function in B0(R0) ⊂ Rn for
some R0 > 1. Assume that Ψ > 0 in B0(1) and that Ψ has a non-degenerate global
maximum at 0. Let ξ2,0∈ M be fixed. We let ig denote the injectivity radius of (M, g)
and for 0 < δ < ig/R0 we let hδ be given by
hδ(x) = Ψ 1 δexp −1 ξ2,0(x) . (1.4)
In particular, hδ is supported in the geodesic ball Bg(ξ2,0, R0δ), and is allowed to change
sign if Ψ changes sign. Remember that (M, g) is said to be of positive Yamabe type if 4g+ cnSg is a positive operator. Our main result states as follows.
Theorem 1.1. Let (M, g) be a closed Riemannian manifold of dimension n ∈ {4, 5} of positive Yamabe type, not conformally diffeomorphic to the standard sphere (Sn, gstd).
Let ξ1,0 and ξ2,0 be distinct points in M and define hδ as in (1.4). Let δ > 0 small be
fixed and let H be any function in the class C(H) defined in (1.7) below. Then, for any 0 < ε ≤ ε0 small enough, there exists a positive solution uε of:
4guε+ cnSg+ hδ+ εHuε= u2
∗−1
in M . This family (uε)0<ε≤ε0 blows-up with finite energy at two distincts simple blow-up
points as ε → 0 and has a zero weak limit.
Note that when (M, g) is of positive Yamabe type and hδ is given by (1.4) the operator
4g+ cnSg+ hδ remains positive for small δ. On the other side, the nonnegativity of
4g+ cnSg+ hδ is a necessary condition to the existence of positive solutions of (1.5)
(see for instance [9], Lemma 2.1).
In the case of the standard sphere, remarkably, an analogue of Theorem 1.1 is available – unlike in the case of perturbations of the Yamabe equation when hδ ≡ 0 as investigated in
[8, 19]. Here the additional assumption that Ψ in (1.4) has negative average compensates for the vanishing of the Riemannian mass.
Theorem 1.2. Let ξ1,0 and ξ2,0 be distinct points in Sn, n ∈ {4, 5}, and define hδ as in
(1.4). Assume in addition that there holds: Z
R4
Ψ(y)dy < 0.
Let δ > 0 small be fixed and let H be any function in the class C(H) defined in (1.7) below. Then, for any 0 < ε ≤ ε0 small enough, there exists a positive solution uε of:
4guε+ n(n − 2) 4 + hδ+ εH uε= u2 ∗−1 ε (1.6)
in M . This family (uε)0<ε≤ε0 blows-up with finite energy at two distincts simple blow-up
points as ε → 0 and has a zero weak limit.
The families (uε)ε of positive solutions that we construct in Theorems 1.1 and 1.2 blow
up as a sum of two isolated simple bubbles of nonequivalent weights. The highest one concentrates at a point ξ2,δ satisfying hδ(ξ2,δ) > 0, while the lowest one concentrates at
ξ1,0. In our constructions, δ is fixed small enough so that hδ(ξ1,0) = 0 by (1.4). To our
knowledge, Theorems 1.1 and 1.2 yield the first example of multi-bubble configurations in dimensions 4 and 5 when the limiting operator 4g+ cnSg+ hδ is positive; clustering
phenomena in the degenerate case had been previously constructed in [27, 29]. The weak limit of our families (uε)εis zero, and this is a necessary condition when n ∈ {4, 5}
by [5].
A few comments on the choice of hδ and H are in order here. First, we point out that
Theorems 1.1 and 1.2 require no smallness assumption on hδ(ξ2,0) = Ψ(0) and therefore
yield existence of (blowing-up) positive solutions for model equations like (1.1) when the limiting potential k is allowed to be much larger than cnSg at a blow-up point. Also, in
Theorem 1.2, no equivariance assumption is needed on hδ. The class C(H) of functions
H considered in Theorems 1.1 and 1.2 is defined as
C(H) = {H ∈ C∞(M ) satisfying (7.7) and (7.8) below }. (1.7) These functions H are used to construct the lowest bubble and can be chosen with great generality. By (1.7) we can choose H ≥ 0, in which case cnSg+ hδ+ εH in (1.5) (and
its counterpart in (1.6)) approaches cnSg+ hδ from above as ε → 0. Remark also that
since we assumed Ψ(0) > 0, the limiting potential always satisfies cnSg+ hδ> cnSg at
in Rn, which is possible in Theorem 1.1, it also satisfies cnSg+ hδ≥ cnSg everywhere in
M . However it does not satisfy cnSg+ hδ > cnSg everywhere in M , in adequation with
the results of [5]. Considering the additional hδ in the potential cnSg+ hδ brings in a
new set of technical problems which are not easily dealt with. They arise in Section 7, where the smallness assumption on δ is quantified and to which we refer for more details. We should also point out that, in some cases, Theorems 1.1 and 1.2 remain true when hδ given by (1.4) is replaced by a suitable smooth function h. Sufficient conditions on h
ensuring this are given in Remark 7.1 below.
Finally, the constructions that we produce here can only occur in dimensions 4 and 5. Indeed, as a consequence of the 3-dimensional sup-inf inequality, solutions of (1.1) can only blow-up as sums of bubbles of comparable weights when n = 3 (see [9], Theorem. 5.2, and see also [10] for other examples of bubbling phenomena in dimension 3). And when n ≥ 7, as a consequence of [5], energy-bounded families (uε)ε of solutions of (1.1)
only exist if limε→0kε = cnSg at all blow-up points (this remains true if n = 6 under
additional assumptions, see [9] prop. 8.1). Let us also mention that the picture when we drop the bounded-energy assumption is radically different: equivariant infinite-energy solutions when k > cnSg and when (Mn, g) is the standard sphere have been constructed
in [4] (when n ≥ 5) and very recently in [31] (when n = 4).
1.2. Strategy of proof of Theorems 1.1 and 1.2. First, we explain how an a priori blow-up analysis yields necessary conditions on the bubbling configuration in our setting. Assume that we are given a family (uε)ε of solutions of (1.5) (or (1.6)) that blows-up
with two bubbles – which are not a priori assumed to be isolated. By the H1-theory of [26] uε writes as
uε = W1,ε+ W2,ε+ o(1) in H1(M ), (1.8)
where Wi,ε, i = 1, 2 are given by (1.3) for some families (µi,ε)ε, i = 1, 2 of positive
numbers going to 0 and for some families (ξ1,ε)ε and (ξ2,ε)ε of points in M converging
towards ξ1 and ξ2 as ε → 0. If we assume now that h(ξ2) > 0, there is not much freedom
left: there necessarily holds that h(ξ1) = 0 and that (µ1,ε)ε and (µ2,ε)ε have to satisfy
µ1,ε= (C1+ o(1))µ2,εln 1 µ2,ε if n = 4, µ31,ε= (C2+ o(1))µ2,ε if n = 5, (1.9)
as ε → 0, for positive constants C1, C2. In particular, ξ1 and ξ2 are distinct and ξ2,ε is
the center of the highest bubble. Similarly, the value of µ1,ε is constrained in terms of ε
by: µ1,εln 1 µ1,ε = (C10 + o(1))ε if n = 4, µ1,ε= (C20 + o(1))ε if n = 5, (1.10)
for positive C10, C20. We refer to Appendix A where relations (1.9) and (1.10) are proven.
finite-dimensional reduction in H1(M ) to produce the constructions of Theorems 1.1 and 1.2 this would force us to work with an extremely high precision, since an expansion of Iε(W1,ε+ W2,ε) involves terms of order µ22,ε, where Iε is the energy functional of (1.5)
(or (1.6)). When n = 5, for instance, µ22,ε is comparable to µ61,ε in view of (1.9), which would force us to estimate the H1(M ) norm of the error in the nonlinear procedure with a precision o(µ31,ε). Finding a suitable ansatz for the approximated bubble W1,ε that
both reaches this precision and comes with explicit estimates to be able to compute the additional contributions in Iε(W1,ε+ W2,ε) seems both unnatural and technically out of
reach.
We overcome this technical difficulty by combining a priori pointwise asymptotic analysis techniques to a nonlinear finite-dimensional procedure in H1(M ). This new approach was recently developed by the first author in [21, 22] to construct instability examples for critical elliptic systems in a coupled supercritical setting. It goes as follows: we first perform the standard nonlinear procedure in H1(M ) and construct a candidate solution W1,ε+ W2,ε+ φε of (1.5) (or (1.6)) up to kernel elements, with φε controlled in
H1(M ). We use here the classical Lyapunov-Schmidt approach that has been developed in the last decades, see for instance [16, 20, 23, 25, 32] and the references therein. Since the H1(M ) bound on φε is not precise enough to proceed as usual, we then obtain a
thorough pointwise decription of the blow-up behavior of φε using techniques in the
spirit of those developed in [7] and [9]. In particular, we do not proceed via an expansion of the reduced-energy in our approach: we conclude our proof by showing that the kernel elements can be annihilated for suitable values of the parameters, and we use for this the latter pointwise estimates on φε.
In view of (1.9), the bubbling configurations that we investigate in this work can be thought of as the low-dimensional counterpart of towering phenomena in higher dimensions. Examples of towering phenomena for positive solutions have recently been constructed in dimensions n ≥ 7 in [17], carrying out a nice improvement of the usual energy methods, but taking advantage of a radial symmetry assumption. In this respect our approach, which relies on a priori analysis methods to perform the finite-dimensional reduction, allows us to overcome the absence of symmetry in the configuration of our bubbles. We believe our method will prove useful in future work when addressing the construction of involved bubbling configurations, for instance in the absence of symmetries.
The structure of the article is as follows. In Section 2 we introduce the bubbling profiles W1,ε and W2,ε. An elevated precision is required on W1,ε while a naive choice of W2,ε is
enough. In Section 3 we apply the standard nonlinear reduction procedure in H1(M ) and construct a solution W1,ε+ W2,ε+ φε of (1.5) up to kernel elements. Sections 4, 5 and 6
are the core of the analysis of the paper. In Section 4 we turn the H1 bound on φεinto a
global C0 one and show that φε = o(W1,ε+ W2,ε) in C0(M ). This requires an adaptation
of the techniques of [7], since W1,ε+ W2,ε+ φε is only a solution of (1.5) up to kernel
elements and can change sign. In Section 5 we improve the global estimate of Section 4 into a sharp higher-order pointwise control on φεaround ξ2. This again involves blow-up
elements. On one side, those pertaining to the kernel associated to the lowest bubble W1,ε are simply expanded using energy estimates. On the other side, those coming from
the highest bubble W2,ε cannot be dealt with in this way and are instead computed
using the precise pointwise asymptotics of Section 5. The analysis in Sections 4, 5 and 6 does not use (1.4) and can be performed in full generality. Section 7 contains the concluding vanishing argument in the proof of Theorems 1.1 and 1.2. Finally, Appendix A describes the a priori analysis considerations leading to (1.9).
Acknowledgments: The authors warmly thank Olivier Druet and Emmanuel Hebey for stimulating discussions and valuable comments on the manuscript.
2. Notations and bubbling profiles
Let (M, g) be a closed Riemannian manifold of dimension n ∈ {4, 5} of positive Yamabe type – that is, such that 4g+ cnSg is coercive, where cn= 4(n−1)n−2 and Sg is the scalar
curvature of (M, g). By the standard conformal normal coordinates result of Lee-Parker [12], there exists Λ ∈ C∞(M × M ) such that by letting Λξ= Λ(ξ, ·) there holds that:
Λξ(ξ) = 1, ∇Λξ(ξ) = 0, (2.1) that Sgξ(ξ) = 0, ∇Sgξ(ξ) = 0, 4gξSgξ(ξ) = 1 6|Wg(ξ)| 2 g, (2.2)
where Sgξ denotes the scalar curvature of the conformal metric gξ= Λ 4 n−2
ξ g, and that,
for any point ξ ∈ M there holds for arbitrarily large given N : expgξξ ∗ gξ (y) = 1 + O(|y| N), (2.3)
C1-uniformly in ξ ∈ M and in y ∈ TξM , |y| ≤ C. Here exp gξ
ξ denotes the exponential
map for the metric gξ at ξ with the identification of TξM to Rnvia a smooth orthonormal
basis of TξM defined in an open set containing ξ. For any ξ ∈ M , we let Ggξ denote the
Green’s function of the operator 4gξ+ cnSgξ in M . Since n ∈ {4, 5}, the result of [12]
asserts that for any ξ ∈ M one has:
Ggξ(ξ, exp
gξ
ξ (y)) =
1 (n − 2)ωn−1
|y|2−n+ A(ξ) + O(|y|) (2.4)
as |y| → 0, where ωn−1 is the volume of the standard sphere Sn−1. The constant A(ξ) in
(2.4) is called the mass of Ggξ at ξ. It smoothly depends on ξ and there holds A(ξ) > 0
for any ξ ∈ M provided (M, g) is not conformally diffeomorphic to the standard sphere, and A ≡ 0 otherwise. For the sake of clarity we also recall the conformal covariance property of the conformal laplacian: for any v ∈ C∞(M ) and ξ ∈ M ,
4g+ cnSg (Λξv) = Λ2 ∗−1 ξ 4gξ+ cnSgξ (v).
If Gg denotes the Green’s function of 4g+ cnSg in M this yields in particular that :
Let ξ1,0 and ξ2,0 be distinct points of M , and let r0 > 0 be such that
8r0 < min ig(M ), dg(ξ1,0, ξ2,0), inf
ξ∈Mdgξ(ξ1,0, ξ2,0), (2.6)
where ig denotes the injectivity radius of (M, g) and dg and dgξ respectively denote
the Riemannian distance associated to the metric g and gξ. Let H and h be smooth
functions in M . Assume that H is supported in Bgξ1,0(ξ1,0, 2r0), where Bgξ1,0 denotes
the geodesic ball with respect to the metric gξ1,0, that 4g+ cnSg+ h is coercive and that
h is supported in M \Bgξ1,0(ξ1,0, 2r0), so that the supports of h and H are disjoint. The
precise form of H and h will only come into play in Section 7, and we do not assume for now that (1.4) holds. The blow-up analysis performed in Sections 4 and 5 and the expansions in Section 6 will only rely on the assumption on their supports. Similarly, whether (M, g) is conformally diffeomorphic to the standard sphere or not only comes into play in Section 7.
Let µ1 > 0 and ξ1∈ M . Following [8] we define, for x ∈ M :
ˆ W1,µ1,ξ1(x) = (n − 2)ωn−1Ggξ1(ξ1, x)Λξ1(x)× dgξ1(ξ1, x)n−2µ n−2 2 1 µ21+dgξ1(ξ1, x) 2 n(n − 2) 1−n2 if dgξ1(ξ1, x) < r0, rn−20 µ n−2 2 1 µ21+ r 2 0 n(n − 2) 1−n2 if dgξ1(ξ1, x) ≥ r0. (2.7)
For µ1 > 0 and ξ1∈ M , let T1,µ1,ξ1 be the unique solution in M of:
4g+ cnSg+ h
T1,t1,ξ1 = −h ˆW1,µ1,ξ1. (2.8)
It is a smooth function in M since h is supported in M \Bgξ1,0(ξ1,0, 2r0). Let χ ∈ C∞(R+)
be a smooth nonnegative function, with χ ≡ 1 in [0, r0] and χ ≡ 0 in [2r0, +∞). Define,
for µ1, µ2 > 0, ξ1, ξ2∈ M , and for x ∈ M :
W1,µ1,ξ1(x) = ˆW1,µ1,ξ1(x) + T1,µ1,ξ1(x), W2,µ2,ξ2(x) = χ(dg(ξ2, x))µ n−2 2 2 µ22+ dg(ξ2, x)2 n(n − 2) 1−n2 . (2.9)
As announced in the introduction, the choice of W2,µ2,ξ2 is rougher than the choice of
W1,µ1,ξ1; in particular, the conformal correction at ξ2 is not required. Note also that
kernel elements, for 1 ≤ j ≤ n and x ∈ M : Z1,0,µ1,ξ1(x) = (n − 2)ωn−1dgξ1(ξ1, x) n−2G gξ1(ξ1, x)χ dgξ1(ξ1, x) Λξ1(x) × µ n−2 2 1 dgξ1(ξ1, x)2 n(n − 2) − µ 2 1 ! µ21+dgξ1(ξ1, x) 2 n(n − 2) !−n2 , Z1,j,µ1,ξ1(x) = (n − 2)ωn−1dgξ1(ξ1, x) n−2G gξ1(ξ1, x)χ dgξ1(ξ1, x) Λξ1(x) × µ n 2 1 expgξξ1 1 −1 (x), ej(ξ1) gξ1(ξ1) µ21+dgξ1(ξ1, x) 2 n(n − 2) !−n2 , Z2,0,µ2,ξ2 = χ (dg(ξ2, x)) µ n−2 2 2 dg(ξ2, x)2 n(n − 2) − µ 2 2 µ22+dg(ξ2, x) 2 n(n − 2) −n2 , Z2,j,µ2,ξ2 = χ (dg(ξ2, x)) µ n 2 2 D expξ2−1 (x), ej(ξ2) E gξ2(ξ2) µ22+dg(ξ2, x) 2 n(n − 2) −n2 . (2.10) In (2.10) we denoted by the same notation (e1(y), · · · , en(y)) two families of orthonormal
vector fields, respectively for gξ1 and g, defined in open sets containing respectively ξ1
and ξ2.
We conclude this subsection with a remark. Let ξ1∈ Bgξ1,0(ξ1,0, r0). By (2.7), and since
h(y) = 0 for any dgξ1(ξ1, y) ≤ r0 by (2.6), T1,µ1,ξ1 in (2.8) is represented, with (2.1), (2.5)
and (2.7), as: T1,µ1,ξ1(x) = −(n − 2)ωn−1 n(n − 2) n−22 µ n−2 2 1 Z M
Gh(x, y)h(y)Gg(y, ξ1)dvg(y)
+O(µ
n+2 2
1 ),
(2.11)
where Gh denotes the Green’s function of 4g+ cnSg+ h in M and the O(µ
n+2 2
1 ) term is
in C2(M ) and is independent of the choice of µ1 and ξ1. Similarly we also obtain that,
for any y ∈ M \{ξ1}:
Gg(ξ1, x) = Gh(ξ1, x) +
Z
M
Gg(ξ1, y)h(y)Gh(y, x)dvg(y). (2.12)
The latter with and (2.7), (2.9) and (2.11) shows in particular that, for dgξ1(ξ1, x) ≥ r0,
we have: W1,µ1,ξ1 = (n − 2)ωn−1(n(n − 2)) n−2 2 µ n−2 2 1 Gh(ξ1, ·) + O(µ n+2 2 1 ) in C2(M ). (2.13) 3. Reduced problem in H1(M ) Let ε > 0 and let t1, t2 be positive numbers. We define:
As explained in the introduction, this choice of µ1,ε and µ2,ε is not a lucky guess but is
necessary and driven by conditions (1.9) and (1.10). The blowing-up solutions of (1.5) and (1.6) that we construct in this paper are bubbles modeled on (2.9) for the choice of µ1, µ2 given by (3.1). For t1, t2 > 0 and ξ1, ξ2 ∈ M we thus let, for 0 ≤ j ≤ n:
W1,ε,t1,ξ1 = W1,µ1,ε(t1),ξ1,
W2,ε,t1,t2,ξ2 = W2,µ2,ε(t1,t2),ξ2,
Z1,ε,j,t1,ξ1 = Z1,j,µ1,ε(t1),ξ1,
Z2,ε,j,t1,t2,ξ2 = Z2,µ2,ε(t1,t2),ξ2,
(3.2)
where µ1,ε(t1) and µ2,ε(t1, t2) are given by (3.1). Let A0 be a connected compact set in
(0, +∞), and let:
A = A0× Bgξ1,0(ξ1,0, r0) × A0× Bg(ξ2,0, r0). (3.3)
Throughout the paper, for the sake of clarity and since no confusion will occur, whenever (t1,ε, ξ1,ε, t2,ε, ξ2,ε)εwill denote a family of points in A, the families µ1,ε(t1,ε), µ2,ε(t1,ε, t2,ε),
W1,ε,t1,ε,ξ1,ε, Z1,j,ε,t1,ξ1,ε, W2,ε,t1,ε,t2,ε,ξ2,ε, Z2,j,ε,t1,ε,t2,ε,ξ2,ε, 0 ≤ j ≤ n will just be
de-noted by µ1, µ2, W1, . . . . Similarly, (t1,ε, ξ1,ε, t2,ε, ξ2,ε)ε will often simply be denoted by
(t1, ξ1, t2, ξ2). Given (t1,ε, ξ1,ε, t2,ε, ξ2,ε)ε, and adopting these notations, we will also let,
for any x ∈ M :
θ1(x) = µ1+ dgξ1(ξ1, x) and θ2(x) = µ2+ dg(ξ2, x). (3.4)
The points ξ1 and ξ2 will be thought of as the centers, respectively, of the lowest and
the highest bubble. By the choice of A in (3.3), they will always satisfy dgξ1,0(ξ1,0, ξ1) ≤ r0 and dg(ξ2,0, ξ2) ≤ r0,
so that by (2.6) the supports of Z1,j or H are disjoint from the supports of W2, Z2,k or
h, for 0 ≤ j, k ≤ n. In particular, W2 is supported in the region where h is nontrivial.
As a first step of our proof, we apply the standard finite-dimensional reduction scheme to this family of bubbling profiles. For any ε > 0 and (t1, ξ1, t2, ξ2) ∈ A, where A is as
in (3.3), let
Kε,t1,ξ1,t2,ξ2 = Span{Z1,j, Z2,k, 0 ≤ j, k ≤ n}, (3.5)
where the Zi,j are defined in (3.2), and let Kε,t⊥1,ξ1,t2,ξ2 be its orthogonal for the scalar
product:
hu, vi = Z
M
h∇u, ∇vi + (cnSg+ h + εH)uv
dvg. (3.6)
In the following, all the H1(M )-norms appearing, denoted by k · kH1(M ), will be taken
with respect to this scalar product. Also, throughout this paper, if (fε)ε, (gε)ε denote
families of numbers or functions, the notation “fε . gε” will be used to denote the
existence of a positive constant C independent of ε such that fε ≤ Cgε for any ε small
enough. If gε≥ 0, we will also write “fε= O(gε)” to say that |fε| . gε.
Proposition 3.1. There exists ε0 > 0 such that for any 0 < ε ≤ ε0 and for any
(t1, ξ1, t2, ξ2) ∈ A, there exists φε(t1, ξ1, t2, ξ2) ∈ Kε,t⊥1,ξ1,t2,ξ2 such that
ΠK⊥ ε,t1,ξ1,t2,ξ2 " W1+ W2+ φε(t1, ξ1, t2, ξ2) −4g+ cnSg+ h + εH −1 W1+ W2+ φε(t1, ξ1, t2, ξ2) 2∗−1 + # = 0, (3.7) where W1and W2are as in (3.2) and where ΠK⊥
ε,t1,ξ1,t2,ξ2 denotes the orthogonal projection
on Kε,t⊥
1,ξ1,t2,ξ2 for (3.6) . In addition, for any 0 < ε ≤ ε0, φε ∈ C
0(A, H1(M ))
and there exists a positive constant C such that, for any 0 < ε ≤ ε0 and for any
(t1,ε, ξ1,ε, t2,ε, ξ2,ε)ε∈ A there holds:
kφε(t1,ε, ξ1,ε, t2,ε, ξ2,ε)kH1(M ) ≤ Cεµ n−2
2
1 , (3.8)
where µ1 is given by (3.1) for t1= t1,ε. Also, φε(t1, ξ1, t2, ξ2) is the unique solution of
(3.7) in Kε,t⊥
1,ξ1,t2,ξ2 satisfying (3.8).
In (3.7) we have let, for any u ∈ H1(M ), u+ = max(u, 0).
Proof. The existence, continuity and uniqueness properties of φε for 0 < ε ≤ ε0 for some
ε0 > 0, as well as (3.7), are a consequence of the general framework developed in [25]
(Proposition 5.1), in which (1.5) and (1.6) fall. The result of [25] generalizes previous ideas developed in [16, 20]. It remains to prove (3.8). Let (t1,ε, ξ1,ε, t2,ε, ξ2,ε)0<ε≤ε0 ∈ A. We
claim that the following estimate holds: there exists a positive constant C, independent on ε and on the choice of the family (t1,ε, ξ1,ε, t2,ε, ξ2,ε)ε, such that for any 0 < ε ≤ ε0,
W1+ W2− 4g+ cnSg+ h + εH −1 W1+ W2 2∗−1 H1(M ) ≤ Cεµ n−2 2 1 , (3.9)
where we used again the notations W1, W2, µ1, µ2, ξ1, ξ2 as above. First, a simple test
function computation using (3.2), together with Sobolev and trace inequalities shows that W1+ W2− 4g+ cnSg+ h + εH −1 W1+ W2 2∗−1 H1(M ) . (4g+ cnSg+ h + εH) (W1+ W2) − (W1+ W2) 2∗−1 Ln+22n (M ) + ||∂inW1+ ∂outW1|| L 2(n−1) n (∂Bgξ 1(ξ1,r0)) ,
where ∂inW1 and ∂outW1 denote the derivative with respect to the unit outward and
and H are disjoint we write, in M \∂Bgξ1(ξ1, r0), that: 4g+cnSg+ h + εH(W1+ W2) − (W1+ W2)2 ∗−1 = 4g+ cnSg+ εH ˆ W1− ˆW2 ∗−1 1 + εHT1+ ˆW2 ∗−1 1 − Wˆ1+ T1 2∗−1 + 4g+ cnSg+ hW2− W2 ∗−1 2 + W12∗−1+ W22∗−1− (W1+ W2)2 ∗−1 . (3.10) On one side, straightforward computations using (2.11) and (3.2) give:
εHT1+ ˆW 2∗−1 1 − Wˆ1+ T1 2∗−1 + W12∗−1+ W22∗−1− (W1+ W2)2 ∗−1 Ln+22n (M ) . εµ n−2 2 1 + µn−21 + (µ1µ2) n−2 2 .
On the other side, straightforward computations give that there holds:
|∂inW1+ ∂outW1| . µ
n+2 2
1 in ∂Bgξ1(ξ1, r0)
and that, both in Bgξ1(ξ1, r0) and in M \Bgξ1(ξ1, r0), there holds:
4g+ cnSg+ εH ˆ W1− ˆW2 ∗−1 1 .µ n+2 2 1 r n−4 1 µ21+ r 2 1 n(n − 2) −n2 , (3.11)
where we have let r1= dgξ1(ξ1, ·) (see for instance [8], Proposition 2.2). As a consequence:
4g+ cnSg+ εH ˆ W1− ˆW2 ∗−1 1 Ln+22n (M )+||∂inW1+ ∂outW1||L2(n−1)n (∂Bgξ 1(ξ1,r0)) . εµ n−2 2 1 .
Finally, straightforward computations using (3.2) show that there holds, for any x ∈ M : 4g+ cnSg+ hW2(x) − W 2∗−1 2 (x) .µ n−2 2 2 θ2(x)2−n, (3.12)
where θ2 is defined in (3.4). This gives in the end:
4g+ cnSg+ hW2(x) − W 2∗−1 2 (x) Ln+22n (M ). µ n−2 2 2 .
Combining all these computations into (3.10) and using the explicit expression of µ1, µ2
given by (3.1) concludes the proof of (3.9). Estimate (3.8) then follows from (3.9) by
the result of [25].
4. C0-theory and uniform a priori pointwise estimates on φε
Let ε0 be given by Proposition 3.1. For 0 < ε ≤ ε0, let (t1, ξ1, t2, ξ2) ∈ A and let
φε= φε(t1, ξ1, t2, ξ2) be given by Proposition 3.1. Equation (3.7) shows that there exist
λεi,j = λεi,j(t1, ξ1, t2, ξ2), i = 1, 2, 0 ≤ j ≤ n, such that φε satisfies:
4g+ cnSg+ h + εH uε= (uε)2 ∗−1 + + X i,j λεi,j4g+ cnSg+ h + εH Zi,j, (4.1)
where we have let:
and W1 = W1,ε,t1,ξ1, . . . are given by (3.2). Since φε ∈ K
⊥
ε,t1,ξ1,t2,ξ2, integrating (4.1)
against Zi,j for 1 ≤ i ≤ 2 and 0 ≤ j ≤ n and using (3.8) yields the existence of a positive
constant C such that for any 0 < ε ≤ ε0 and for any (t1,ε, ξ1,ε, t2,ε, ξ2,ε)ε≤ε0 ∈ A there
holds, for all 1 ≤ i ≤ 2 and 0 ≤ j ≤ n:
|λε
i,j| = |λεi,j(t1,ε, ξ1,ε, t2,ε, ξ2,ε)| ≤ Cεµ
n−2 2
1 . (4.3)
We aim at constructing a solution of (1.5) and (1.6) via (4.1) by finding an element (t1,ε, ξ1,ε, t2,ε, ξ2,ε)ε which annihilates all the λεi,j. This goes through an asymptotic
expansion in C0(A) of the λεi,j as ε → 0, where A is given in (3.3). However, as explained in the introduction, having h > 0 in the region where the center of the highest bubble W2 is expected to be localized requires these expansions to be carried out with a high
precision that cannot be reached with the mere H1 estimate (3.8). In this section we therefore obtain a priori global pointwise asymptotic estimates on φε. These will be
refined into sharp second-order estimates on φε around ξ2 in the next section.
We show that φε is, in a pointwise sense, globally small compared to W1 and W2:
Proposition 4.1. There exists ε1 > 0 and a family of positive numbers (νε)0<ε≤ε1 with
limε→0νε= 0 such that there holds, for any 0 < ε ≤ ε1 and for any (t1, ξ1, t2, ξ2) ∈ A:
φε(t1, ξ1, t2, ξ2)(x) ≤ νε W1(x) + W2(x) for any x ∈ M. (4.4)
Here again W1 and W2 are given by (3.2) and A is as in (3.3). In particular, up to
assuming that ε1 is small enough, we will assume that νε≤ 12 for 0 < ε ≤ ε1.
Proof of Proposition 4. The proof of Proposition 4 is divided into two Lemmas. The first one establishes, for a fixed ε, continuity properties of the mapping φε in strong
spaces.
Lemma 4.2. There exists ε1> 0 such that, for any 0 < ε ≤ ε1 the mappings:
(t1, ξ1, t2, ξ2) ∈ A 7−→ W1,ε,t1,ξ1 + W2,ε,t1,t2,ξ2 ∈ C
0(M )
(t1, ξ1, t2, ξ2) ∈ A 7−→ φε(t1, ξ1, t2, ξ2) ∈ C0(M )
are well-defined and continuous.
Proof. For the first map, the assertion simply follows from the explicit expression of the right-hand side given by (3.2) and by the regularity properties of Λξ1. We thus prove
the Lemma for the second map. First, by (3.1) and (3.8) we let ε1 > 0 be such that, for
any 0 < ε ≤ ε1 and any (t1, ξ1, t2, ξ2) ∈ A there holds:
kφε(t1, ξ1, t2, ξ2)kH1(M ) <
1 2K
−n−22
n , (4.5)
where we have let:
Kn= s 4 n(n − 2)ω 2 n n (4.6)
and ωn is the volume of the standard unit n-sphere. Let 0 < ε ≤ ε1 be fixed and
(4.1) and by an adaptation of Trudinger’s argument [30] (see also [9], Theorem 2.15) we get that
(W1+ W2+ φε)+∈ Ls(M )
for some s > 2∗. Then, with (4.1), a bootstrap procedure applies and shows that uε∈ C2(M ), and hence that φε∈ C0(M ). Note however that φε is not smooth on the
sphere {dgξ1,0(ξ1,0, y) = r0} since ˆW1 is not.
Let now (t1,k, ξ1,k, t2,k, ξ2,k)k≥1be a sequence of points of A converging towards (t1,0, ξ1,0, t2,0, ξ2,0)
and let φk = φε(t1,k, ξ1,k, t2,k, ξ2,k) for any k ≥ 1 and φ0 = φε(t1,0, ξ1,0, t2,0, ξ2,0). By
Proposition 3.1, φk→ φ0 in H1(M ) as k → +∞. Assume first that the sequence (φk)k
is uniformly (in k) bounded in L∞(M ). Then (3.2), (4.3) and standard elliptic theory in (4.1) show that every subsequence of (φk)k admits a subsequence which converges in
C0(M ), and therefore to φ0. In this case, thus, φk→ φ0 in C0(M ) as k → +∞.
We therefore assume that, up to a subsequence, kφkkL∞(M ) → +∞ as k → +∞. A
Green’s representation formula for uk given by (4.2) with (4.1) and standard properties
of Green’s functions (see [24]) show, since (W1 + W2+ φk)+ ≥ 0, that there exists a
positive constant Cε, independent of k, such that:
inf
M φk ≥ −Cε (4.7)
for any k ≥ 1. In particular, we might as well assume that maxM(φk)+ → +∞ as
k → +∞ and let xkbe such that φk(xk) = kφkkL∞(M )= maxM(φk)+→ +∞ as k → +∞.
We let µk= φk(xk)−
2
n−2 and, for any x ∈ B0(ig(M )/µk), we let gk = exp∗
xkg(µk·) and ˜ uk(x) = µ n−2 2 k uk(expxk(µkx)).
With (4.1), ˜uk satisfies, for any y ∈ B0(ig(M )/µk):
4gku˜k(y) + µ2k cnSg+ h + εH (yk)˜uk(y) = (˜uk(y))2 ∗−1 + + µ n+2 2 k X i,j λεi,j(t1,k, ξ1,k, t2,k, ξ2,k) 4g+ cnSg+ h + εH Zi,j(yk),
where we have let yk = expxk(µky). By (4.7), by the definition of xk and since ε is fixed
throughout this proof there holds:
−Cεµ n−2 2 k ≤ ˜uk(y) ≤ 1 + C0µ n−2 2 k
for some positive constant C0 and for any y ∈ B0(ig(M )/µk). By (4.3) and standard
elliptic theory, ˜uk converges therefore in Cloc1 (Rn) to ˜u0, with 0 ≤ ˜u0 ≤ 1, solution of
4ξu˜0 = ˜u2
∗−1
0 .
By the definition of xk we also have ˜u0(0) = 1, so that the classification result in [3]
implies that k˜u0kL2∗(Rn)= K
−n−22
holds thus: Z Bxk(Rµk) |φk|2∗dvg ≥ Z Bxk(Rµk) |uk|2∗dvg+ O(µ n−2 2 k ) = Z B0(R) |˜uk|2 ∗ dvgk+ O(µ n−2 2 k ) = Z B0(R) |˜u0|2 ∗ dx + o(1) = (1 + εR)Kn−n+ o(1),
as k → +∞, where limR→+∞εR = 0. This is a contradiction with (4.5) for R and k
large enough and concludes the proof of the Claim.
The second Lemma establishes a rough version of (4.4):
Lemma 4.3. Let ε1 be as in Lemma 4.2 and let (t1,ε, ξ1,ε, t2,ε, ξ2,ε)0<ε≤ε1 be a family of
points of A. Define, for any 0 < ε ≤ ε1:
νε:= φε(t1,ε, ξ1,ε, t2,ε, ξ2,ε) W1,ε,t1,ε,ξ1,ε+ W2,ε,t1,ε,t2,ε,ξ2,ε C0(M ) . Then νε→ 0 as ε → 0.
Proof. We prove Lemma 4.3 by contradiction, and therefore assume the existence of a sequence (εk)k, 0 < εk≤ ε1, with εk→ 0 as k → +∞, such that
φk W1+ W2 C0(M ) ≥ η0 (4.8)
for some η0 > 0, for all k ≥ 1. In (4.8), for the sake of simplicity and using the
previous notations, we simply wrote t1,εk = t1, ξ1,εk = ξ1, W1,εk,t1,εk,ξ1,εk = W1,
φεk(t1,k, ξ1,k, t2,k, ξ2,k) = φk and so on. We will keep these notations throughout the
proof of the Lemma and it will be implicit that we will be working with the quantities given by (3.2), associated to the sequences (εk)k and (t1,εk, ξ1,εk, t2,εk, ξ2,εk)k.
The proof of Lemma 4.3 consists in an asymptotic a priori analysis of the sequence (φk)k and is divided into several steps.
Step 1: local convergence. We first show that, for 1 ≤ i ≤ 2, there holds, up to a subsequence: µ n−2 2 i uk expξi(µi·) → U0 in C 1 loc(Rn), (4.9)
as k → +∞, where uk= uεk is as in (4.2), where µ1, µ2 are given by (3.1) and
U0(x) = 1 + |x| 2 n(n − 2) 1−n2 for x ∈ Rn. (4.10)
vi,k(x) = µ
n−2 2
i uk expξi(µix). Letting gi,k = exp
∗
ξig(µi·), with (4.1) vi,k satisfies, for
any x ∈ B0(ig(M )/µi) and for xk = expξi(µix):
4gi,kvi,k(x) + µ 2 i cnSg+ h + εkH(xk)vi,k = (vi,k)2 ∗−1 + +X i,j λεk i,jµ n+2 2 i 4g+ cnSg+ h + εkH Zi,j(xk). (4.11)
We show that vi,k is uniformly bounded in Cloc0 (Rn) by investigating its positive and
negative part separately. First, a straightforward adaptation of the arguments in [18] shows, with (4.11), that (vi,k)+= max(vi,k, 0) satisfies the following equation in a weak
sense:
4gi,k(vi,k)++ µ
2
i cnSg+ h + εkH(expξi(µi·))(vi,k)+ ≤ (vi,k)
2∗−1 + +X i,j λεk i,jµ n+2 2 i 4g+ cnSg+ h + εkH
Zi,j(expξi(µi·))1vi,k>0.
(4.12) By (4.3) and (2.10) we have: X i,j λεk i,jµ n+2 2 i 4g+ cnSg+ h + εkH Zi,j(xk) → 0 in Cloc0 (Rn) as k → +∞. Also, µ2i cnSg+ h + εkH
expξi(µi·) → 0 in Cloc0 (Rn), and by the definition of vi,k, by
(3.8) and (4.2) there holds that:
lim
r→0lim supk→+∞
Z
Bx(r)
vi,k2∗(y)dy = 0 ∀x ∈ Rn.
Hence, an adaptation of Trudinger’s argument [30] to (4.12) shows that for any R > 0 there exists CR> 0 such that
k(vi,k)+kC0(B
0(R)) ≤ CR (4.13)
for k large enough.
Independently, let Gkdenote the Green’s function of 4g+ (cnSg+ h + εkH) in M and let
(xk)k be a sequence of points in M . By (3.2), (4.1) and (4.3) a representation formula
for uk gives:
uk(xk) & −εkµ
n−2 2
1 W1(xk) + W2(xk).
We used here that by (2.10) and (3.2) there exists a positive constant C depending only on n such that |Zi,j| ≤ CWi for i = 1, 2 and j = 0, . . . , n. Since by (2.6) and (3.3) there
holds lim infk→+∞dg(ξ1, ξ2) > 0, the latter inequality shows in particular that for any
x ∈ B0(ig(M )/µi):
vi,k & −εk− εkµn−21 µ
n−2 2
2 .
This shows that:
(vi,k)−→ 0 in Cloc0 (Rn) as k → +∞. (4.14)
Standard elliptic theory, with (4.13) and (4.14), shows with (4.11) that vi,k converges in
Step 2: uniform lower bound on φk. We now show that there exists a positive
sequence ηk→ 0 as k → +∞ such that, up to a subsequence,
φk(x) & −ηk W1+ W2(x) for any x ∈ M. (4.15)
Let (xk)k a sequence of points such that
φk W1+ W2 (xk) = inf x∈M φk W1+ W2 . (4.16)
Remember that W1+ W2 is positive in M . We write again a representation formula for
uk with (4.1) and (4.3), which gives:
W1+ W2+ φk (xk) & Z M Gk(xk, y) W1+ W2+ φk 2∗−1 + dvg − εkµ n−2 2 1 W1+ W2 & −εkµ n−2 2 1 W1+ W2 + Z Bξ1(Rkµ1) Gε(xk, y) W1+ W2+ φk 2∗−1 + dvg + Z Bξ2(Rkµ2) Gk(xk, y) W1+ W2+ φk 2∗−1 + dvg, (4.17) where Rk > 0 is chosen so that Rkµi → 0 as k → +∞ for i = 1, 2 and such that
Bξ1(Rkµ1) and Bξ2(Rkµ2) are disjoint for all k. The integrals in (4.17) are estimated
with Fatou’s lemma and (4.9) which in turn, with (4.16), yields (4.15).
Step 3: Blow-up analysis. Step 2 shows in particular that, for k large enough,
uk= W1+ W2+ φk≥ 1 2 W1+ W2 in M. So uk actually solves, in M 4g+ cnSg+ h + εkH uk= u2 ∗−1 k + X i,j λεk i,j 4g+ cnSg+ h + εkH Zi,j.
Using (4.3), an adaptation of the blow-up analysis performed in [22] (Proposition 4.1, Steps 4, 5, 6), see also [28], shows that there holds, for any sequence (xk)k of points of
M : |φk(xk)| = o W1(xk) + W2(xk) .
Applying the latter to the sequence (xk)k that achieves the maximum point of W1φ+Wk 2
in M then yields a contradiction with (4.8), and concludes the proof of Lemma 4.3. We now conclude the proof of Proposition 4.1. Let ε1 be as in Lemma 4.2. Then, again
by Lemma 4.2, for any 0 < ε ≤ ε1 there exists (t1,ε, ξ1,ε, t2,ε, ξ2,ε) ∈ A such that:
where A is as in (3.3). Let, for any 0 < ε ≤ ε1, νε be given by Lemma 4.3 for this
maximal family (t1,ε, ξ1,ε, t2,ε, ξ2,ε)0<ε≤ε1. Then, for any x ∈ M , for any 0 < ε ≤ ε1 and
for any (t1, ξ1, t2, ξ2) ∈ A, there holds by (4.18) that:
φε(t1, ξ1, t2, ξ2)(x) W1,ε,t1,ξ1(x) + W2,ε,t1,t2,ξ2(x) ≤ νε.
Since limε→0νε= 0, this proves (4.4) and concludes the proof of Proposition 4.1.
The estimates on |φε(t1, ξ1, t2, ξ2))| given by Proposition 4.1 are, for a given ε, uniform
in the choice of (t1, ξ1, t2, ξ2). This is an important property of our analysis that will be
crucial in the final argument of the proof of Theorems 1.1 and 1.2.
A consequence of Proposition 4.1 is that for any 0 < ε ≤ ε1 and for any (t1, ξ1, t2, ξ2) ∈ A,
we now have
W1+ W2+ φε(t1, ξ1, t2, ξ2) ≥
1
2(W1+ W2) .
In particular, with (4.1), we now see that for any 0 < ε ≤ ε1 and for any (t1, ξ1, t2, ξ2),
uε given by (4.2) actually satisfies in M :
4g+ cnSg+ h + εH uε= u2 ∗−1 ε + X i,j λεi,j4g+ cnSg+ h + εH Zi,j. (4.19)
5. Second-order pointwise estimates
In this section we refine the pointwise estimate on φε given by Proposition 4.1 in balls
of fixed radius centered at ξ2. These improved pointwise estimates will compensate for
the insufficient precision of (3.8) and will be the crucial ingredient of the asymptotic expansion of the λεi,j in Section 6. Let:
A1= A0× Bgξ1,0(ξ1,0, r0) × A0× Bg ξ2,0,
r0
2, (5.1)
where A0 is the compact set in (0, +∞) appearing in (3.3). The second-order estimates
that we obtain are as follows:
Proposition 5.1. There exists ε2 > 0 and C > 0 such that, for any 0 < ε ≤ ε2 and for
any family (t1,ε, ξ1,ε, t2,ε, ξ2,ε)0<ε≤ε2 ∈ A1, where A1 is defined in (5.1), we have:
• If n = 4, and for any x ∈ Bg(ξ2,ε,r20):
|φε(x)| ≤ C νε µ1+ µ2 + µ2|ln θ2(x)| + µ1µ2 ln θ2(x) µ2 µ2 θ2(x)2 ! , (5.2)
• If n = 5, and for any x ∈ Bg(ξ2,ε,r20):
|φε(x)| ≤ C νε µ 3 2 1 + µ 3 2 2 + µ 3 2 2θ2(x) −1+ µ72 1 + µ 3 2 1| ln µ2|µ 3 2 2 µ2 θ2(x)2 32 ! . (5.3)
Here, as before, we have let φε= φε(t1,ε, ξ1,ε, t2,ε, ξ2,ε), µ1 and µ2 are given by (3.1), θ2
is as in (3.4) and νε is given by Proposition 4.1.
Lemma 5.2. Let ε1 and (νε)0<ε≤ε1 be given by Proposition 4.1 and let (t1,ε, ξ1,ε, t2,ε, ξ2,ε)0<ε≤ε1
be a family in A, where A is in (3.3). There exists 0 < ε2≤ ε1 and C > 0 such that:
• If n = 4, and for any x ∈ Bg(ξ2,ε, 2r0):
|φε(x)| ≤ C νε µ1+ µ2 + µ2|ln θ2(x)| + µ1µ2 ln θ2(x) µ2 µ2 θ2(x)2 ! , (5.4)
• If n = 5, and for any x ∈ Bg(ξ2,ε, 2r0):
|φε(x)| ≤ C νε µ 3 2 1 + µ 3 2 2 + µ 3 2 2θ2(x)−1+ µ 7 2 1 + µ 3 2 1| ln µ2|µ 3 2 2 µ2 θ2(x)2 32 ! , (5.5)
where we used the same notations as in the statement of Proposition 5.1.
Proof. Let (t1,ε, ξ1,ε, t2,ε, ξ2,ε)0<ε≤ε1 ∈ A. Throughout the rest of this proof C will denote
a positive constant independent of ε, which might change from one line to another. We will adopt the same notations as before.
We first assume that n = 4. By (4.19) and since the supports of H and W2 are
disjoint, φε = φε(t1,ε, ξ1,ε, t2,ε, ξ2,ε) satisfies in M : 4g+1 6Sg+ h + εH φε− X i,j λεi,jZi,j = W1+ W2+ φε 3 −W1+ W2 3 − 4g+ 1 6Sg+ h + εH W1− W13 + 3W12W2+ 3W1W22 − 4g+1 6Sg+ h W2− W23 . (5.6) For any 0 < ε ≤ ε1, let Gε be the Green’s function of 4g + 16Sg + h with Dirichlet
boundary condition on Bg(ξ2, 2r0) (remember that ξ2 = ξ2,ε). Let (xε)ε≤ε1 be a family
of points in Bg(ξ2, 2r0). For any 0 < ε ≤ ε1, if xε∈ Bg(ξ2, 2r0)\Bg(ξ2, r0), there holds
trivially:
|φε(xε)| ≤ kφεkC0(2r
0\r0), (5.7)
where we have let kφεkC0(2r
0\r0) = kφεkC0(Bg(ξ2,2r0)\Bg(ξ2,r0)). Otherwise we write a
representation formula on Bg(ξ2, 2r0) for φεwith (5.6). This is possible since by definition
W1 is smooth in Bg(ξ2, 2r0) and therefore so is φε. Since the Z1,j and H are supported
outside of Bξ2(2r0) we get that:
where I1 = Z Bg(ξ2,2r0) Gε(xε, y) W1+ W2+ φε 3 (y) −W1+ W2 3 (y) dvg(y), I2 = Z Bg(ξ2,2r0) Gε(xε, y) 4g+1 6Sg+ h W1− W13 dvg(y), I3 = Z Bg(ξ2,2r0) Gε(xε, y) W12W2+ W1W22 dvg(y), I4 = Z Bg(ξ2,2r0) Gε(xε, y) 4g+1 6Sg+ h W2− W23 dvg(y).
First, by (3.12) and standard properties of Green functions (see [24]) there holds:
I4≤ Cµ2|ln θ2(xε)| , (5.9)
where θ2 is as in (3.4). Then, by (3.2) and since by (2.6) there holds dgξ1(ξ1, y) > r0 for
any y ∈ Bξ2(2r0), it is easily seen that there holds:
I2 ≤ Cµ31. (5.10)
Straightforward computations using (3.2) show that there holds:
I3 ≤ C µ2|ln θ2(xε)| + µ1µ2 ln θ2(xε) µ2 µ2 θ2(xε)2 . (5.11)
Finally, Proposition 4.1 shows that W1+ W2+ φε 3 −W1+ W2 3 ≤ CW1+ W2 2 |φε|,
so that straightforward computations lead to:
I1≤ C µ21+ µ2 ln θ2(xε) µ2 µ2 θ2(xε)2 kφεkC0(2r 0), (5.12)
where we have let kφεkC0(2r
0)= kφεkC0(Bξ2(2r0)). Combining (5.9)–(5.12) in (5.8) gives,
with (5.7) and (3.1), that there holds, for any (xε)0<ε≤ε1, xε∈ Bg(ξ2, 2r0):
|φε(xε)− 4 X j=0 λε2,jZ2,j(xε)| ≤ C kφεkC0(2r 0\r0)+ |λ ε 2,0|µ2+ 4 X j=1 |λε2,j|µ22+ µ2|ln θ2(xε)| + µ21+ µ2 ln θ2(xε) µ2 µ2 θ2(xε)2 kφεkC0(2r 0)+ µ1µ2 ln θ2(xε) µ2 µ2 θ2(xε)2 ! . (5.13) By evaluating (5.13) at suitable points satisfying dg(xε, ξ2) ≤ µ2 one gets with (3.1) the
following estimate on the λε2,j, 0 ≤ j ≤ 4:
4
X
j=0
|λε2,j| ≤ Cµ2 kφεkC0(2r
so that, using Proposition 4.1, (5.13) improves into: |φε(xε)| ≤ C νε µ1+ µ2 + µ2|ln θ2(xε)| + µ21+ µ2 ln θ2(xε) µ2 µ2 θ2(xε)2 kφεkC0(2r 0)+ µ1µ2 ln θ2(xε) µ2 µ2 θ2(xε)2 ! . (5.15) We now claim that the following result holds true:
Claim 5.3. There exists 0 < ε2 ≤ ε1 and C > 0 such that, for any 0 < ε ≤ ε2, there
holds:
kφεkC0(2r
0)≤ Cµ1. (5.16)
Proof. Remember that µ1 = µ1,ε(t1,ε) is given by (3.1). We proceed by contradiction
and assume that for some sequence (εk)k of positive numbers, εk→ 0 as k → +∞, there
holds
kφkkC0(2r
0) µ1 as k → +∞. (5.17)
As before, until the end of this Claim it will be implicit that all the quantities µ1, ξ1, . . .
depend on this subsequence (εk)k according to (3.1) and (3.2). We will let in particular
φk= φεk(t1, ξ1, t2, ξ2). Let (yk)k be a sequence of points such that |φk(yk)| = kφkkC0(2r0).
By (3.1) and (5.17) there holds
µ2|ln θ2(yk)| ≤ µ2| ln µ2| = o(kφkkC0(2r 0))
as k → +∞, and by Proposition 4.1 limk→+∞νεk = 0 so that (5.15) and (5.17) show
that
θ2(yk) ≤ Cµ2, (5.18)
where θ2 is defined in (3.4). For any y ∈ B0(µr02) we then let:
˜ φk(y) = 1 kφkkC0(2r 0) φk expξ2(µ2y). (5.19)
On one hand, (5.15) and (5.17) show that:
| ˜φk(y)| ≤ C ln(1 + |y|) (1 + |y|)2 + o(1) on B0 r0 µ2 . (5.20)
On the other hand, (3.2), (3.11), (3.12) and (5.17) show that:
µ22 kφkkC0(2r 0) − 4g+1 6Sg+ h W1− W13 + 3W12W2+ 3W1W22 − 4g+ 1 6Sg+ h W2− W23 ! expξ2(µ2·)
˜ φ0 as k → +∞, where ˜φ0 solves in R4: 4euclφ˜0 = 3U02φ˜0+ 4 X j=0 ˜ λ02,jVj. (5.21)
Here eucl denotes the Euclidean metric, U0 is as in (4.10) and for any 0 ≤ j ≤ 4 we have
let: ˜ λ02,j = lim k→+∞ λεk 2,j µ2kφkkC0(2r 0) .
That this limit exists, up to a subsequence, is a consequence of (5.14) and (5.17). Also, in (5.21), Vj is defined in Rn for any n ≥ 3 by:
V0(y) = |y|2 n(n − 2)− 1 1 + |y| 2 n(n − 2) −n2 , Vj(y) = yj 1 + |y| 2 n(n − 2) −n2 , for any 1 ≤ j ≤ n. (5.22)
Passing (5.20) to the limit shows that ˜φ0 ∈ L4(R4) so that integrating (5.21) against Vj
shows that ˜λ0
2,j = 0 for any 0 ≤ j ≤ 4. Now the result of [1] implies that:
˜
φ0 ∈ Vect{Vj, 0 ≤ j ≤ 4}.
To conclude the proof of Claim 5.3 we now prove that ˜φ0∈ Vect{Vj, 0 ≤ j ≤ 4}⊥, where
the orthogonal is taken for the usual scalar product in ˙H1(R4). This will imply that ˜
φ0 ≡ 0, which is a contradiction since | ˜φ0(˜y0)| = 1 by (5.18), where ˜y0 is the limit of 1
µ2 exp
−1
ξ2 (yk) as k → +∞.
To prove that ˜φ0∈ Vect{Vj, 0 ≤ j ≤ 4}⊥, we write that by Proposition 3.1 there holds,
for any 0 ≤ j ≤ 4:
hφk, Zji = 0,
where h·, ·i is the scalar product given by (3.6). Let now R > 0 be fixed. The latter equality implies, since Z2,j, 0 ≤ j ≤ n, is supported in Bg(ξ2, 2r0), that:
Z Bg(ξ2,Rµ2) h∇φk, ∇Z2,ji + 1 6Sg+ hφkZ2,j ! dvg= Z ∂Bg(ξ2,Rµ2) ∂νZ2,jφkdσg − Z Bg(ξ2,2r0)\Bg(ξ2,Rµ2) 4g+1 6Sg+ h Z2,jφkdvg. (5.23)
Straightforward computations using (2.10) show that there holds, for y ∈ Bg(ξ2, 2r0):
4gZ2,0(y) − 3W2(y)2Z2,0(y)
≤ Cµ2θ2(y)−2,
4gZ2,j(y) − 3W2(y)2Z2,j(y)
≤ Cµ22θ2(y)−3, for 1 ≤ j ≤ 4,
(5.24)
where θ2 is as in (3.4). Now, (3.2) and (5.15) show that:
that Z Bg(ξ2,2r0)\Bg(ξ2,Rµ2) µ2θ2(·)−2φkdvg = o(µ2kφkkC0(2r 0)), and that Z Bg(ξ2,2r0)\Bg(ξ2,Rµ2) W22Z2,jφkdvg ≤ C (1 + R)3 + o(1) µ2kφkkC0(2r 0), so that (5.23) becomes: Z Bg(ξ2,Rµ2) h∇φk, ∇Z2,ji + 1 6Sg+ hφkZ2,j ! dvg ≤ Cln(1 + R) 1 + R + o(1) µ2kφkkC0(2r 0).
Dividing both sides by µ2kφkkC0(2r
0), using the definition of ˜φk in (5.19), letting first k
go to +∞ and then R → +∞ we obtain that:
Z
R4
h∇ ˜φ0, ∇Vjidx = 0.
This proves that ˜φ0 ∈ Vect{Vj, 0 ≤ j ≤ 4}⊥and, as explained above, gives a contradiction,
thus concluding the proof of Claim 5.3.
Plugging (5.16) into (5.15) and using (3.1) concludes the proof of Lemma 5.2 for n = 4.
Assume now that n = 5. The proof is similar to the four-dimensional case. By (4.19) φε satisfies in M : 4g+ 3 16Sg+ h + εH φε− X i,j λεi,jZi,j = W1+ W2+ φε 73 −W1+ W2 73 +W1+ W2 73 − W 7 3 1 − W 7 3 2 − 4g+ 3 16Sg+ h + εH W1− W 7 3 1 − 4g+ 3 16Sg+ h W2− W 7 3 2 . (5.25) For any 0 < ε ≤ ε1, let Gε be the Green’s function of 4g + 163Sg + h with Dirichlet
boundary condition on Bg(ξ2, 2r0). Let (xε)0<ε≤ε1 be any family of points in Bg(ξ2, 2r0).
For any 0 < ε ≤ ε1, if xε∈ Bg(ξ2, 2r0)\Bg(ξ2, r0), there holds trivially:
|φε(xε)| ≤ kφεkC0(2r 0\r0).
Otherwise we write a representation formula for φε with (5.25). Since the Z1,j are
using (4.3), we get that there holds, for any (xε)0<ε≤ε1, xε ∈ Bg(ξ2, 2r0): |φε(xε) − 5 X j=0 λε2,jZ2,j(xε)| ≤ C kφεkC0(2r 0\r0)+ µ 7 2 1 + µ 3 2 2θ2(xε)−1+ µ 3 2 1 µ2 θ2(xε) 2 + µ21+ µ2 θ2(xε) 2! kφεkC0(2r 0) ! . (5.26) Evaluating again the latter estimate at suitable points satisfying dg(xε, ξ2,ε) ≤ µ2 one
gets, with (3.1), the following estimate:
5 X j=0 |λε 2,j| ≤ Cµ 3 2 2 kφεkC0(2r 0)+ µ 3 2 1 , (5.27)
so that, using (3.1) and Proposition 4.1, (5.26) improves into:
|φε(xε)| ≤ C νε µ 3 2 1 + µ 3 2 2 + µ 7 2 1 + µ 3 2 2θ2(xε)−1+ µ 3 2 1 µ2 θ2(xε) 2 + µ21+ µ2 θ2(xε) 2! kφεkC0(2r 0) ! . (5.28)
As before, we prove the following claim:
Claim 5.4. There exists 0 < ε2 ≤ ε1 and C > 0 such that, for any 0 < ε ≤ ε2, there
holds: kφεkC0(2r 0)≤ Cµ 3 2 1. (5.29)
Proof. Here again we proceed by contradiction and assume that for some sequence (εk)k
of positive numbers, εk→ 0 as k → +∞, there holds
kφkkC0(2r 0) µ
3 2
1 as k → +∞, (5.30)
using the same notations as in the proof of Claim 5.3. Let (yk)k be a sequence of points
such that |φk(yk)| = kφkkC0(2r
0). By (3.1), (3.4) and (5.30) there holds µ 3 2
2θ2(yk)−1 =
o(kφkkC0(2r
0)) so that (5.28) and (5.30) show that
θ2(yk) ≤ Cµ2. (5.31)
For any y ∈ B0(µr02) we let again:
˜ φk(y) = 1 kφkkC0(2r 0) φk expξ2(µ2y). (5.32)
Then (5.28) and (5.30) show that there holds:
| ˜φk(y)| ≤ C
1
(1 + |y|)2 + o(1) for any y ∈ B0
r0
µ2
As before, (3.11), (3.12), Proposition 4.1, (5.25), (5.30) and standard elliptic theory show that ˜φk converges, up to a subsequence, in Cloc1 (R5) towards ˜φ0 as k → +∞, where
˜ φ0 solves in R5: 4euclφ˜0= 7 3U 4 3 0φ˜0+ 5 X j=0 ˜ λ02,jVj, (5.34)
where eucl is the Euclidean metric, U0 is as in (4.10), the Vj are as in (5.22) and for any
0 ≤ j ≤ 5 we have let: ˜ λ02,j = lim k→+∞ λεk 2,j µ 3 2 2kφkkC0(2r 0) ,
which exists, up to a subsequence, by (5.27) and (5.30). Passing (5.33) to the limit shows that ˜φ0 ∈ L
10
3 (R5) so that integrating (5.34) against Vj shows first that ˜λ0
2,j = 0
for any 0 ≤ j ≤ 5 and then, by the result of [1], that: ˜
φ0 ∈ Vect{Vj, 0 ≤ j ≤ 5}. (5.35)
As before, we now prove that ˜φ0 ∈ Vect{Vj, 0 ≤ j ≤ 5}⊥. By Proposition 3.1 there holds
again
hφk, Zji = 0
for any 0 ≤ j ≤ 5, where h·, ·i is given by (3.6), so that for any R > 0 this implies that:
Z Bg(ξ2,Rµ2) h∇φk, ∇Z2,ji + 3 16Sg+ hφkZ2,j ! dvg= Z ∂Bg(ξ2,Rµ2) ∂νZ2,jφkdσg − Z Bg(ξ2,2r0)\Bg(ξ2,Rµ2) 4g+ 3 16Sg+ h Z2,jφkdvg. (5.36)
Straightforward computations using (2.10) show that there holds, for y ∈ Bξ2(2r0):
4gZ2,0(y) − 7 3W2(y) 4 3Z2,0(y) ≤ Cµ 3 2 2θ2(y)−3, 4gZ2,j(y) − 7 3W2(y) 4 3Z2,j(y) ≤ Cµ 5 2 2θ2(y)−4, for 1 ≤ j ≤ 5, (5.37)
where θ2 is as in (3.4). Now, (3.2), (5.28) and (5.30) show that:
so that (5.36) becomes: Z Bg(ξ2,Rµ2) h∇φk, ∇Z2,ji + 3 16Sg+ hφkZ2,j ! dvg ≤ C 1 + R + o(1) µ 3 2 2kφkkC0(2r 0).
Dividing both sides by µ
3 2
2kφkkC0(2r
0), letting first the k → +∞ and then R → +∞ gives
as before that ˜φ0 ∈ Vect{Vj, 0 ≤ j ≤ 5}⊥ and hence ˜φ0 ≡ 0. But this is a contradiction
with (5.35), since | ˜φ0(˜y0)| = 1 by (5.31), where ˜y0 is the limit of µ12 exp−1ξ2 (yk), and
concludes the proof of Claim 5.4.
Now, plugging (5.29) into (5.28) yields:
|φε(xε)| ≤ C νε µ 3 2 1 + µ 3 2 2 + µ 7 2 1 + µ 3 2 2θ2(xε) −1 + µ 3 2 1 µ2 θ2(xε) 2! .
Writing down again a representation formula for (5.25) and using the latter to estimate the term involving φε then concludes the proof of Lemma 5.2 for n = 5.
Note that the precision that we reach in Lemma 5.2 is related to the nature of φε, in
particular to the property of φε to be orthogonal to the kernel elements.
End of the proof of Proposition 5.1. Of course the constants ε2 and C given by Lemma
5.2 do depend on the choice of the family (t1,ε, ξ1,ε, t2,ε, ξ2,ε)ε≤ε1. To conclude the proof
of Proposition 5.1 we establish as before their uniformity. We only write the argument for n = 4 since the n = 5 case works identically. First, the right-hand side of (5.2) (seen as a continuous function in M ) is obviously continuous in (t1, ξ1, t2, ξ2). Therefore,
by Lemma 4.2, there exists a family (t1,ε, ξ1,ε, t2,ε, ξ2,ε)0<ε≤ε1 ∈ A such that for any
0 < ε ≤ ε1 there holds: φε(t1,ε, ξ1,ε, t2,ε, ξ2,ε) νε(µ1,ε+ µ2,ε) + µ2,ε|ln θ2(·)| + µ1,εµ2,ε ln θ 2,ε(·) µ2 µ2 θ2(·)2 C0(B g(ξ2,0,r0)) = sup (t1,ξ1,t2,ξ2)∈A φε(t1, ξ1, t2, ξ2) νε(µ1+ µ2) + µ2|ln θ2(·)| + µ1µ2 ln θ 2(·) µ2 µ2 θ2(·)2 C0(B g(ξ2,0,r0)) , (5.38) where ξ2,0 and r0 are as in (2.6), νε is given by Proposition 4.1 and A is given by (3.3).
Note in particular that ξ2,ε∈ Bg(ξ2,0, r0). Let ε2 and C be the constants associated to
this family (t1,ε, ξ1,ε, t2,ε, ξ2,ε)0<ε≤ε1 by Lemma 5.2. Let now ξ2 ∈ Bg(ξ2,0,
r0
2). Then:
Bg(ξ2,
r0
Therefore, for any 0 < ε ≤ ε2, sup (t1,ξ1,t2,ξ2)∈A1 φε(t1, ξ1, t2, ξ2) νε(µ1+ µ2) + µ2|ln θ2(·)| + µ1µ2 ln θ 2(·) µ2 µ2 θ2(·)2 C0(Bg(ξ 2,r02)) ≤ sup (t1,ξ1,t2,ξ2)∈A1 φε(t1, ξ1, t2, ξ2) νε(µ1+ µ2) + µ2|ln θ2(·)| + µ1µ2 ln θ 2(·) µ2 µ2 θ2(·)2 C0(B g(ξ2,0,r0)) ≤ C,
where in the last inequality we used that A1⊂ A and Bg(ξ2,0, r0) ⊂ Bg(ξ2,ε, 2r0) in order
to apply (5.38) and Lemma 5.2. This concludes the proof of (5.2). 6. Asymptotic Expansion along the kernel
For any 0 < ε ≤ ε2 and (t1, ξ1, t2, ξ2) ∈ A1 we let φε = φε(t1, ξ1, t2, ξ2) be given by
Proposition 3.1, where A1 is as in (5.1) and ε2 is given by Proposition 5.1. In this
Section we obtain an asymptotic expansion of the functions λεi,j, 1 ≤ i ≤ 2, 0 ≤ j ≤ n defined in (4.19). Throughout this section, all the asymptotic expansions that we will write hold in C0(A1).
6.1. Expansion of the λε1,j, 0 ≤ j ≤ n. We first obtain an asymptotic expansion of the λε1,j, 0 ≤ j ≤ n:
Lemma 6.1. The following expansions hold in C0(A1) as ε → 0, where A1 is as in
(5.1): If n = 4: k∇V0k2L2(R4)λε1,0(t1, ξ1, t2, ξ2) = 2C1(4)H(ξ1)t1− C2(4)A(ξ1) + C3(4) 2 Fh(ξ1) e−2t1ε + o(e− 2t1 ε ), k∇Vjk2L2(R4)λε1,j(t1, ξ1, t2, ξ2) = 4 C1(4)∇jH(ξ1)t1− C2(4)∇jA(ξ1) + C3(4) ∇jFh(ξ1) e−3t1ε + o(e−3t1ε ). (6.1) If n = 5: k∇V0k2L2(R5)λε1,0(t1, ξ1, t2, ξ2) = 4 3C1(5)H(ξ1)t 2 1− 2C2(5)t31A(ξ1) + 2C3(5)t31Fh(ξ1) ε3+ o(ε3), k∇Vjk2L2(R5)λε1,j(t1, ξ1, t2, ξ2) = 5 C1(5)∇jH(ξ1)t31− C2(5)∇jA(ξ1)t41+ C3(5) ∇jFh(ξ1)t41 ε4+ o(ε4). (6.2)
In (6.1) and (6.2) the Vj are defined in (5.22), A(ξ1) denotes the mass of the Green’s
given by
Fh(ξ) =
Z
M
Gh(ξ, y)h(y)Gg(y, ξ1)dvg(y), (6.3)
C1(n), C2(n) are positive constants given by (6.13), and C3(n) is a positive constant
defined by (6.10) below.
The explicit values of the constants Ci(n) do not come into play in our final argument.
It is important to notice that the term A(ξ1) − Fh(ξ1) is just, in view of (2.12), the
mass of the Green’s function Gh(ξ1, ·) at ξ1 by analogy with (2.4). Note that the mass
of Gh(ξ1, ·) at ξ1 exists because h is supported in M \Bgξ1,0(ξ1,0, 2r0). If h were just a
smooth function in M the next order term in expansion (2.4) would likely be singular too, as explained in [24].
Proof. These expansions only require the H1 estimate on φε given by (3.8). By (3.2),
the Z1,j, 0 ≤ j ≤ n are supported in Bgξ1,0(ξ1,0, 2r0). Since W2, the Z2,j and h are
supported in M \Bgξ1,0(ξ1,0, 2r0) it is easily seen that (4.19) rewrites in Bgξ1,0(ξ1,0, 2r0)
as: n X j=0 λε1,j(t1, ξ1, t2, ξ2) 4g+ cnSg+ εH Z1,j = εHT1 + 4g+ cnSg+ εH ˆ W1+ φε − ˆW1+ φε 2∗−1 −h Wˆ1+ T1+ φε 2∗−1 − Wˆ1+ φε 2∗−1i , (6.4)
where T1 is given by (2.8) with µ1 given by (3.1). To estimate the λε1,j we integrate (6.4)
against Z1,j. First, by (2.11), (3.1) and (3.2) there holds, for 0 ≤ j ≤ n:
ε Z M HT1Z1,jdvg = ( o(e−2t1ε ) if n = 4 o(ε3) if n = 5, if j = 0, ( o(e−3t1ε ) if n = 4 o(ε4) if n = 5, if 1 ≤ j ≤ n. (6.5)
Then, we write that Wˆ1+T1+ φε 2∗−1 − Wˆ1+ φε 2∗−1 − (2∗− 1) ˆW12∗−2T1 . ˆW1+ φε 2∗−3 |T1|2+ |T 1|2 ∗−1 + ( ˆW1+ φε)2 ∗−2 − ˆW12∗−2 |T1| . |φε|2 ∗−2 + ˆW12∗−3|φε||T1| + Wˆ2 ∗−3 1 + |φε|2 ∗−3 |T1|2+ |T1|2 ∗−1 . (6.6)
Straightforward computations with (2.11), (3.1) and (3.2) give that:
while (2.11), (3.1), (3.2) and (3.8) give, with H¨older’s inequality: Z M |φε|2∗−2|T1| + ˆW12∗−3|φε||T1| + |φε|2∗−3|T1|2|Z1,j|dvg = ( o(e−2t1ε ) if n = 4 o(ε3) if n = 5, if j = 0, ( o(e−3t1ε ) if n = 4 o(ε4) if n = 5, if 1 ≤ j ≤ n.
Combining the latter computations in (6.6) then gives:
− Z M h ˆ W1+ T1+ φε 2∗−1 − Wˆ1+ φε 2∗−1i Z1,jdvg = − (2∗− 1) Z M ˆ W12∗−2T1Z1,jdvg+ ( o(e−2t1ε ) if n = 4 o(ε3) if n = 5, if j = 0, ( o(e−3t1ε ) if n = 4 o(ε4) if n = 5, if 1 ≤ j ≤ n. (6.7)
With (2.11) and (3.2) it is now easily seen that there holds:
− (2∗− 1) Z M ˆ W12∗−2T1Z1,0dvg = C3(n)µn−21 Fh(ξ1) + o(µn−21 ) (6.8) and − (2∗− 1) Z M ˆ W12∗−2T1Z1,jdvg = nC3(n)µn−11 ∇jFh(ξ1) + o(µn−11 ), (6.9)
where µ1 is given by (3.1) and where C3(n) > 0 is given by the following expansion as
ε → 0: − Z M ˆ W12∗−1T1dvg = C3(n)µn−21 Fh(ξ1) + O(µn1), (6.10)
for Fh defined in (6.3). Finally, by (2.7), (3.1), (3.8) and since by construction φε is
orthogonal to the Z1,j, 0 ≤ j ≤ n, an adaptation of the computations in [8] (Section 6)
and Z M h 4g+ cnSg+ εH ˆ W1+ φε − ˆW1+ φε 2∗−1i Z1,jdvg = 4 C1(4)∇jH(ξ1)t1− C2(4)∇jA(ξ1) e−3t1ε + o(e− 3t1 ε ) if n = 4, 5C1(5)∇jH(ξ1)t13− C2(5)∇jA(ξ1)t41 ε4+ o(ε4) if n = 5, (6.12) where C1(n) and C2(n) are two positive constants defined by the following expansion as
ε → 0: 1 2 Z M |∇ ˆW1|2g+ (cnSg+ εH) ˆW12dvg− 1 2∗ Z M ˆ W12∗dvg = 1 nK −n n + C1(4)H(ξ1)t1− C2(4)A(ξ1) e−2t1ε + o(e−2t1ε ) if n = 4 C1(5)H(ξ1)t21− C2(5)A(ξ1)t31 ε3+ o(ε3) if n = 5. (6.13)
It remains to notice that by (3.2) there holds:
hZ1,j, Z1,ki = δjkk∇Vjk2L2(Rn)+ o(1)
for 0 ≤ j, k ≤ n, and even:
hZ1,0, Z1,ji =
(
o(e−t1ε) if n = 4
o(ε) if n = 5,
for 1 ≤ j ≤ n, in C0(A1) as ε → 0, where h·, ·i is given by (3.6) and Vj is as in (5.22).
With (3.1), (6.4), (6.5), (6.7), (6.8), (6.9), (6.11) and (6.12) this concludes the proof of
the Lemma.
6.2. Expansion of the λε2,j, 0 ≤ j ≤ n. In this subsection we obtain an asymptotic expansion of the λε2,j. Unlike the case of the λε1,j, the expansion of the λε2,j now crucially relies on the precise pointwise asymptotics on φε obtained in Sections 4 and 5.
Lemma 6.2. The following expansions hold in C0(A1) as ε → 0, where A1 is as in
If n = 5: k∇V0k2L2(R5)λε2,0(t1, ξ1, t2, ξ2) = 4 3D1(5)h(ξ2)t 2 2− D2(5)(t1t2) 3 2Gh(ξ1, ξ2) ε6+ o(ε6), k∇Vjk2L2(R5)λε2,j(t1, ξ1, t2, ξ2) = 5D1(5)∇jh(ξ2)t32− D2(5)t 3 2 1t 5 2 2∇jGh(ξ1, ξ2) ε9+ o(ε9). (6.15)
In (6.14) and (6.15) the Vj are as in (5.22), Gh denotes the Green’s function of 4g+
cnSg+ h in M and its derivative is taken with respect to ξ2. Also, D1(n) and D2(n) are
positive constants given by (6.20) and (6.23) below.
Proof. As before, all the asymptotic expansions that we will write here take place in C0(A1). By (2.7), (2.8), (3.2) and (4.19) and since the Z1,j and H vanish on Bg(ξ2, 2r0),
for any 0 < ε ≤ ε2 and for any (t1, ξ1, t2, ξ2) ∈ A1 there holds:
n X j=0 λε2,j(t1, ξ1, t2, ξ2) 4g+ cnSg+ h Z2,j = 4g+ cnSg ˆW1− ˆW2 ∗−1 1 + ˆW12∗−1− Wˆ1+ T1 2∗−1 +4g+ cnSg+ h W2− W2 ∗−1 2 − (2∗− 1)W2∗−2 2 W1 −h(W1+ W2)2 ∗−1 − W12∗−1− W22∗−1− (2∗− 1)W22∗−2W1 i −h W1+ W2+ φε 2∗−1 − (W1+ W2)2 ∗−1 − (2∗− 1)(W1+ W2)2 ∗−2 φε i + 4g+ cnSg+ h φε− (2∗− 1)W2 ∗−2 2 φε − (2∗− 1)h W1+ W2 2∗−2 − W22∗−2iφε. (6.16)
We integrate (6.16) against Z2,j for 0 ≤ j ≤ n. First, using (2.7), (2.11), (3.2) and (3.11)
we get that, for any 0 ≤ j ≤ n:
Mimicking the computations that led to (6.8) and (6.9) we get that: Z M " 4g+ cnSg+ h W2− W2 ∗−1 2 # Z2,0dvg = 2D1(4)h(ξ2)εt1t22e− 2t1 ε + o(εe−2t1ε ) if n = 4 4 3D1(5)h(ξ2)ε 6t2 2+ o(ε6) if n = 5 , (6.18) and Z M " 4g+ cnSg+ h W2− W2 ∗−1 2 # Z2,jdvg = ( 4D1(4)∇jh(ξ2)ε2t1t32e− 3t1 ε + o(ε2e−3t1ε ) if n = 4 5D1(5)∇jh(ξ2)ε9t32+ o(ε9) if n = 5 , (6.19)
where the positive constants D1(4) and D1(5) are defined by:
1 2 Z M |∇W2|2g+ (cnSg+ h)W22dvg− 1 2∗ Z M W22∗dvg = 1 nK −n n + ( D1(4)h(ξ2)εt1t22e− 2t1 ε + o(εe−2t1ε ) if n = 4 D1(5)h(ξ2)ε6t22+ o(ε6) if n = 5, (6.20)
where Kn is defined in (4.6). Similarly, direct computations using (2.13), (3.1) and (3.2)
show that − (2∗− 1) Z M W22∗−2W1Z2,0dvg = − D2(4)εt2e− 2t1 ε Gh(ξ1, ξ2) + o(εe−2t1ε ) if n = 4, − D2(5)ε6(t1t2) 3 2Gh(ξ1, ξ2) + o(ε6) if n = 5, (6.21) and, for 0 ≤ j ≤ n, that
−(2∗−1) Z M W22∗−2W1Z2,jdvg = − 4D2(4)ε2t22e−3t1ε ∇jGh(ξ1, ξ2) + o(ε2e−3t1ε ) if n = 4, − 5D2(5)ε9t 3 2 1t 5 2 2∇jGh(ξ1, ξ2) + o(ε9) if n = 5, (6.22) where ∇G(ξ1, ξ2) stands for the derivative of Gh(ξ1, ·) at ξ2. Also, in (6.21) and (6.22),
the positive constants D2(n) are given by the following expansion as ε → 0:
Z M W22∗−1W1dvg = D2(4)εt2e− 2t1 ε Gh(ξ1, ξ2) + o(εe−2t1ε ) if n = 4 D2(5)ε6(t1t2) 3 2Gh(ξ1, ξ2) + o(ε6) if n = 5. (6.23)
and using (2.13) and (3.2) yields: Z M h (W1+ W2)2 ∗−1 − W12∗−1− W22∗−1− (2∗− 1)W22∗−2W1 i Z2,jdvg = ( o(εe−2t1ε ) if n = 4 o(ε6) if n = 5, if j = 0, ( o(ε2e−3t1ε ) if n = 4 o(ε9) if n = 5, if 1 ≤ j ≤ n. (6.24)
We now estimate the components in (6.16) where φε appears:
Claim 6.3. There holds, as ε → 0, in C0(A1) :
Z M h W1+ W2+ φε 2∗−1 − (W1+ W2)2 ∗−1 − (2∗− 1)(W1+ W2)2 ∗−2 φε i Z2,jdvg = ( o(εe−2t1ε ) if n = 4 o(ε6) if n = 5, if j = 0, ( o(ε2e−3t1ε ) if n = 4 o(ε9) if n = 5, if 1 ≤ j ≤ n. (6.25)
Proof. Let 0 < ε ≤ ε2. By Proposition 4.1 we can write that there holds, in M :
W1+ W2+ φε 2∗−1 − (W1+ W2)2 ∗−1 − (2∗− 1)(W1+ W2)2 ∗−2 φε . (W1+ W2)2 ∗−3 |φε|2. (6.26)
On one hand, using (3.1), (3.2), (4.4) and (6.26) we get that:
